Fundamental Rules for Differentiation
Rule (I): Differentiation of a constant function is zero i.e., \(\frac{d}{dx}\left( c \right)=0\).
Rule (II): Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that \(\frac{d}{dx}\left\{ c.f\left( x \right) \right\}=c.\frac{d}{dx}\left( f\left( x \right) \right)\).
This is the derivative of a constant times a function is the constant times the derivative of the function.
Rule (III): If f(x) and g(x) are differentiable functions, then show that f(x) ± g(x) are also differentiable such that \(\frac{d}{dx}\left[ f\left( x \right)\pm g\left( x \right) \right]=\frac{d}{dx}f\left( x \right)\pm \frac{d}{dx}g\left( x \right)\).
That is the derivative of the sum or difference of two functions is the sum or difference of their derivatives.
Rule (IV): If f(x) and g(x) are two differentiable functions, then f(x).g(x) is also differentiable such that \(\frac{d}{dx}\left[ f\left( x \right).g\left( x \right) \right]=f\left( x \right)\frac{d}{dx}\left[ g\left( x \right) \right]+\frac{d}{dx}\left[ f\left( x \right) \right].g\left( x \right)\).
Rule (V) (Quotient rule): If f(x) and g(x) are two differentiable functions and g(x)≠0 then \(\frac{f\left( x \right)}{g\left( x \right)}\) is also differentiable such that \(\frac{d}{dx}\left\{ \frac{f\left( x \right)}{g\left( x \right)} \right\}=\frac{g\left( x \right)\frac{d}{dx}\left[ f\left( x \right) \right]-f\left( x \right).\frac{d}{dx}\left[ g\left( x \right) \right]}{{{\left[ g\left( x \right) \right]}^{2}}}\).
Relation between \(\frac{dy}{dx}\) and \(\frac{dx}{dy}\): Let x and y be two variables connected by a relation of the form f(x, y) = 0 Let Δx be a small change in x and let Δy be the corresponding change in y. Then \(\frac{dy}{dx}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta y}{\Delta x}\) and \(\frac{dx}{dy}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta x}{\Delta y}\).
Now,
\(\frac{\Delta y}{\Delta x}.\frac{\Delta x}{\Delta y}=1\),
\(\underset{\Delta x\to 0}{\mathop{\lim }}\,\left[ \frac{\Delta y}{\Delta x}.\frac{\Delta x}{\Delta y} \right]=1\),
\(\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta y}{\Delta x}.\underset{\Delta y\to 0}{\mathop{\lim }}\,\frac{\Delta x}{\Delta y}=1\) [∵ ∇x → 0 ⇔ Δy → 0]
\(\frac{dy}{dx}.\frac{dx}{dy}=1\),
Hence, \(\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\).
Example: Differentiate f(x) = sinx.logx
Solution: Using product rule of differentiation
\(f’\left( x \right)=\frac{d}{dx}\left( \sin x \right).\log x+\sin x.\frac{d}{dx}\left( \log x \right)\),
\(\,=\log x.\cos x+\frac{\sin x}{x}\).
Example: Find the derivative of \(f\left( x \right)=\frac{\operatorname{logx}}{{{x}^{2}}}\).
Solution: Using quotient rule of differentiation
\(f’\left( x \right)=\frac{{{x}^{2}}\frac{d}{dx}\left( \log x \right)-\log x\frac{d}{dx}\left( {{x}^{2}} \right)}{{{\left( {{x}^{2}} \right)}^{2}}}\),
\(=\frac{{{x}^{2}}.\frac{1}{x}-\log x\left( 2x \right)}{{{x}^{4}}}\),
\(=\frac{x-2x\log x}{{{x}^{4}}}\),
\(=\frac{1-2\log x}{{{x}^{3}}}\).